Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 385324, 16 docx

Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 385324, 16 pages doi:10.1155/2011/385324 Research Article Study of an Approximation Process of Time Optimal Control for Fractional Evolution Systems in Banach Spaces JinRong Wang 1 and Yong Zhou 2 1 Department of Mathematics, Guizhou University, Guiyang, G uizhou 550025, China 2 Department of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, China Correspondence should be addressed to Yong Zhou, yzhou@xtu.edu.cn Received 1 October 2010; Accepted 9 December 2010 Academic Editor: J. J. Trujillo Copyright q 2011 J. Wang and Y. Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The paper is devoted to the study of an approximation process of time optimal control for fractional evolution systems in Banach spaces. We firstly convert time optimal control problem into Meyer problem. By virtue of the properties of the family of solution operators given by us, the existence of optimal controls for Meyer problem is proved. Secondly, we construct a sequence of Meyer problems to successive approximation of the original time optimal control problem. Finally, a new approximation process is established to find the solution of time optimal control problem. Our method is different from the standard method. 1. Introduction It has been shown that the accurate modelling in dynamics of many e ngineering, physics, and economy systems can be obtained by using fractional differential equations. Numerous applications can be found in viscoelasticity, electrochemistry, control, porous media, electromagnetic, and so forth. There has been a great deal of interest in the solutions of fractional differential equations in analytical and numerical sense. One can see the monographs of Kilbas et al. 1, Miller and Ross 2,Podlubny3, and Lakshmikantham et al. 4. The fractional evolution equations in infinite dimensional spaces attract many authors including us see, for instance, 5–21 and the references therein. When the fractional differential equations describe the performance index and system dynamics, a classical optimal control problem reduces to a fractional optimal control problem. The optimal control of a fractional dynamics system is a fractional optimal control with system dynamics defined with partial fractional differential equations. 2AdvancesinDifference Equations There h as been very little work in the area of fractional optimal control pr oblems 18, 22, especially the time optimal control for fractional evolution equations 19. Recalling that the research on time optimal control problems dates back to the 1960s, many problems such as existence and necessary conditions for optimality and controllability have been discussed, for example, see 23 for the finite dimensional case and 7, 24–37 for the infinite dimensional case. Since the cost functional for a time optimal control problem is the infimum of a number set, it is different with the Lagrange problem, the Bolza problem and the Meyer problem, which arise some new difficulties. As a result, we regard the time optimal control as another problem which is not the same as the above three problems. Motivated by our previous work in 18–21, 38, we consider the time optimal control problem P of a fractional evolution system governed by C D q t z  t   Az  t   f  t, z  t  ,B  t  v  t  ,t∈  0,τ  ,q∈  0, 1  , z  0   z 0 ∈ X, v ∈ V ad , 1.1 where C D q t is the Caputo fractional derivative of order q, A : DA → X is the infinitesimal generator of a strongly continuous semigroup {Tt,t ≥ 0}, V ad is the admissible control set and f : I τ :0,τ × X × X → X will be specified latter. Let us mention, we do n ot study the time optimal control problem P of the above system by standard method used in our earlier work 19. In the present paper, we will construct a sequences of Meyer problems P ε n  to successive approximation time optimal control problem P. Therefore, we need introduce the following new fractional evolution system C D q s x  s   k q Ax  s   k q f  ks,x  s  ,B  ks  u  s  ,s∈  0, 1  , x  0   z 0 ∈ X, w   u, k  ∈ W, 1.2 whose controls are taken from a product space W will be specified latter. By applying the family of solution operators T k and S k see Lemma 3.7 associated with the family of C 0 -semigroups with parameters and some probability density functions, the existence of optimal controls f or Meyer problems P ε  is proved. Then, we show that there exists a subsequence of Meyer problems P ε n  whose corresponding sequence of optimal controls {w ε n }∈W converges to a time optimal control of problem P in some sense. In other words, in a limiting process, the sequence {w ε n }∈W can be used to find the solution of time optimal control problem P. The existence of time optimal controls for problem P is proved by this constructive approach which provides a new method to solve the time optimal control. The rest of the paper is organized as follows. In Section 2, some notations and preparation results are given. In Section 3,weformulatethetimeoptimalcontrolproblemP and Meyer problem P ε .InSection 4, the existence of optimal controls for Meyer problems P ε  is proved. Finally, we display the Meyer approximation process of time optimal control and derive the main result of this paper. Advances in Difference Equations 3 2. Preliminaries Throughout this paper, we denote by X a Banach space with the norm ·.Foreachτ< ∞,letI τ ≡ 0,τ and CI τ ,X be the Banach space of continuous functions from I τ to X with the usual supremum norm. Let A : DA → X be the infinitesimal generator of a strongly continuous semigroup {Tt,t ≥ 0}. This means that there exists M>0suchthat sup t∈I τ Tt≤M. We will also use f L p I τ ,R   to denote the L p I τ ,R   norm of f whenever f ∈ L p I τ ,R   for some p with 1 <p<∞. Let us recall the following definitions in 1. Definition 2.1. The fractional integral of order γ with the lower limit zero for a function f is defined as I γ f  t   1 Γ  γ   t 0 f  s   t − s  1−γ ds, t > 0,γ>0, 2.1 provided the right side is pointwise defined on 0, ∞,whereΓ· is the gamma function. Definition 2.2. Riemann-Liouville derivative of order γ with the lower limit zero for a function f : 0, ∞ → R can be written as L D γ f  t   1 Γ  n − γ  d n dt n  t 0 f  s   t − s  γ1−n ds, t > 0,n− 1 <γ <n. 2.2 Definition 2.3. The Caputo derivative of order γ for a function f : 0, ∞ → R can be written as C D γ f  t   L D γ  f  t  − n−1  k0 t k k! f k  0   ,t>0,n− 1 <γ<n. 2.3 Remark 2.4. i If ft ∈ C n 0, ∞,then C D γ f  t   1 Γ  n − γ   t 0 f n  s   t − s  γ1−n ds  I n−γ f n  t  ,t>0,n− 1 <γ<n. 2.4 ii The Caputo derivative of a constant is equal to zero. iii If f is an abstract function with values in X, then integrals which appear in Definitions 2.1 and 2.2 are taken in Bochner’s sense. Lemma 2.5 see 38, Lemma 3.1. If the assumption [A] holds, then 1 for given k ∈ 0,  T, kA is the i nfinitesimal generator of C 0 -semigroup {T k t,t≥ 0} on X, 2 there exist constants C ≥ 1 and ω ∈ −∞, ∞ such that  T k  t  ≤ Ce ωkt , ∀t ≥ 0, 2.5 4AdvancesinDifference Equations 3 if k n → k ε in 0,  T as n →∞, then for arbitrary x ∈ X and t ≥ 0, T k n  t  s −→ T k ε  t  , as n −→ ∞ 2.6 uniformly in t on some closed interval of 0,  T in the strong operator topology sense. 3. System Description and Problem Formulation Consider the following fractional nonlinear controlled system C D q t z  t   Az  t   f  t, z  t  ,B  t  v  t  ,t∈  0,τ  , z  0   z 0 ∈ X, v ∈ V ad . 3.1 We make the following assumptions. A : A is the infinitesimal generator of a C 0 -semigroup {Tt,t ≥ 0} on X with domain DA. F : f : I τ × X × X → X is measurable in t on I τ and for each ρ>0, there exists a constant Lρ > 0 such that for almost all t ∈ I τ and all z 1 ,z 2 ,y 1 ,y 2 ∈ X, satisfying z 1 , z 2 , y 1 , y 2 ≤ρ,wehave   f  t, z 1 ,y 1  − f  t, z 2 ,y 2    ≤ L  ρ   z 1 − z 2     y 1 − y 2    . 3.2 For arbitrary t, z, y ∈ I τ × X × X, there exists a positive constant M>0suchthat   f  t, z, y    ≤ M  1   z     y    . 3.3 B :LetE be a separable reflexive Banach space, B ∈ L ∞ I τ ,LE, X, B ∞ stands for the norm of operator B on Banach space L ∞ I τ ,LE, X. B : L p I τ ,E → L p I τ ,X1 <p<∞ is strongly continuous. U :MultivaluedmapsV· : I τ → 2 E \{Ø} has closed, convex and bounded values. V· is graph measurable and V· ⊆ Ω where Ω is a bounded set of E. Set V ad  { v  ·  | I τ −→ E measurable,v  t  ∈V  t  a.e. } . 3.4 Obviously, V ad /  Ø see 39,Theorem2.1 and V ad ⊂ L p I τ ,E1 <p<∞ is bounded, closed and convex. Based on our previous work 21, Lemma 3.1 and Definition 3.1,weusethefollowing definition of mild solutions for our problem. Advances in Difference Equations 5 Definition 3.1. By the mild solution of system 3.1, we mean that the function x ∈ CI τ ,X which satisfies z  t   T  t  z 0   t 0  t − θ  q−1 S  t − θ  f  θ, z  θ  ,B  θ  v  θ  dθ, t ∈ I τ , 3.5 where T  t    ∞ 0 ξ q  θ  T  t q θ  dθ, S  t   q  ∞ 0 θξ q  θ  T  t q θ  dθ, ξ q  θ   1 q θ −1−1/q  q  θ −1/q  ≥ 0,  q  θ   1 π ∞  n1  −1  n−1 θ −qn−1 Γ  nq  1  n! sin  nπq  ,θ∈  0, ∞  , 3.6 ξ q is a probability density function defined on 0, ∞,thatis ξ q  θ  ≥ 0,θ∈  0, ∞  ,  ∞ 0 ξ q  θ  dθ  1. 3.7 Remark 3.2. i It is not difficult to verify that for v ∈ 0, 1  ∞ 0 θ v ξ q  θ  dθ   ∞ 0 θ −qv  q  θ  dθ  Γ  1  v  Γ  1  qv . 3.8 ii For another suitable definition of mild solutions for fractional differential equations, the reader can refer to 13. Lemma 3.3 see 21, Lemmas 3.2-3.3. The operators T and S have the following properties. i For any fixed t ≥ 0, Tt and St are linear and bounded operators; that is, for any x ∈ X,  T  t  x  ≤ M  x  ,  S  t  x  ≤ qM Γ  1  q  x  . 3.9 ii {Tt,t≥ 0} and {St,t≥ 0} are strongly continuous. We present the following existence and uniqueness of mild solutions for system 3.1. Theorem 3.4. Under the assumptions [A], [B], [F] and [U], for every v ∈ V ad and pq > 1,system 3.1 has a unique mild solution z ∈ CI τ ,X which satisfies the following integral equation z  t   T  t  z 0   t 0  t − θ  q−1 S  t − θ  f  θ, z  θ  ,B  θ  v  θ  dθ. 3.10 6AdvancesinDifference Equations Proof. Consider the ball given by B  {x ∈ C0,T 1 ,X |xt − x 0 ≤1, 0 ≤ t ≤ T 1 },where T 1 would be chosen, and xt≤1  x 0   ρ,0≤ t ≤ T 1 , B⊆C0,T 1 ,X is a closed convex set. Define a map H on B given by  Hz  t   T  t  z 0   t 0  t − θ  q−1 S  t − θ  f  θ, z  θ  ,B  θ  v  θ  dθ. 3.11 Note that by the properties of T and S, assumptions A, F, B,andU,bystandard process see 19,Theorem3.2, one can verify that H is a contraction map on B with T 1 > 0. This means that system 3.1 has a unique mild solution on 0 ,T 1 . Again, using the singular version Gronwall inequality, we can obtain the a prior estimate of the mild solutions of system 3.1 and present the global existence of the mild solutions. Definition 3.5 admissible trajectory.Taketwopointsz 0 , z 1 in the state space X.Letz 0 be the initial state and let z 1 be the desired terminal state with z 0 /  z 1 ,denotezv ≡{zt, v ∈ X | t ≥ 0} be the state trajectory corresponding to the control v ∈ V ad . A trajectory zv is said to be admissible if z0,vz 0 and zt, vz 1 for some finite t>0. Set V 0  {v ∈ V ad | zv is an admissible trajectory}⊂V ad . For given z 0 ,z 1 ∈ X and z 0 /  z 1 ,ifV 0 /  Ø i.e., there exists at least one control from the admissible class that takes the system from the given initial state z 0 to the desired ta rget state z 1 in the finite time.,wesay the system 3.1 can be controlled. Let τv ≡ inf{t ≥ 0 | zt, vz 1 } denote the transition time corresponding to the control v ∈ V 0 /  Øanddefineτ ∗  inf{τv ≥ 0 | v ∈ V 0 }. Then, the time optimal control problem can be stated as follows. Problem Problem P.Taketwopointsz 0 , z 1 in the state space X.Letz 0 be the initial state and let z 1 be the desired terminal state with z 0 /  z 1 . Suppose that there exists at least one control from the admissible class that takes the system from the given initial state z 0 to the desired target state z 1 in the finite time. The time optimal control problem is to find a control v ∗ ∈ V 0 such that τ  v ∗   τ ∗  inf { τ  v  ≥ 0 | v ∈ V 0 } . 3.12 For fixed v ∈ V ad ,  T  τv > 0. Now, we introduce the following linear transformation t  ks, 0 ≤ s ≤ 1,k∈  0,  T  . 3.13 Through this transformation, system 3.1 can be replaced by C D q s x  s   k q Ax  s   k q f  ks,x  s  ,B  ks  u  s  ,s∈  0, 1  , x  0   z  0   z 0 ∈ X, w   u, k  ∈ W, 3.14 Advances in Difference Equations 7 where x·zk·, u·vk·,anddefine W    u, k  | u  s   v  ks  , 0 ≤ s ≤ 1,v∈ V ad ,k∈  0,  T  . 3.15 By Theorem 3.4, one can obtain the following existence result. Theorem 3.6. Under the assumptions of Theorem 3.4, for every w ∈ W and pq > 1,system3.14 has a unique mild solution x ∈ C0, 1,X which satisfies the following integral equation x  s   T k  s  z 0   s 0  s − θ  q−1 S k  s − θ  kf  kθ, x  θ  ,B  kθ  u  θ  dθ, 3.16 where T k  s    ∞ 0 ξ q  θ  T k q  s q θ  dθ, S k  s   q  ∞ 0 θξ q  θ  T k q  s q θ  dθ, 3.17 and {T k q t,t≥ 0} is a C 0 -semigroup generated by the infinitesimal generator k q A. By Lemmas 2.5 and 3.3,itisnotdifficult to verify the following result. Lemma 3.7. The family of solution operators T k and S k given by 3.17 has the following properties. i For any x ∈ X, t ≥ 0, there exists a constant C k q > 0 such that  T k  t  x  ≤ C k q  x  ,  S k  t  x  ≤ qC k q Γ  1  q  x  . 3.18 ii {T k t,t≥ 0} and {S k t,t≥ 0} are also strongly c ontinuous. iii If k q n → k q ε in 0,  T as n →∞, then for arbitrary x ∈ X and t ≥ 0 T k q n  t  s −→ T k q ε  t  , as n −→ ∞ , S k q n  t  s −→ S k q ε  t  , as n −→ ∞ 3.19 uniformly in t on some closed interval of 0,  T in the strong operator topology sense. For system 3.14, we turn to consider the following Meyer problem. 8AdvancesinDifference Equations Meyer Problem P ε  Minimize the cost functional given by J ε  w   1 2ε  x  w  1  − z 1  2  k 3.20 over W,wherexw is the mild solution of 3.14 corresponding to control w,thatis,finda control w ε u ε ,k ε  such that the cost functional J ε w attains its minimum on W at w ε . 4. Existence of Optimal Controls for Meyer Problem P ε  In this section, we discuss the existence of optimal controls for Meyer problem P ε . We show that Meyer problem P ε  has a solution w ε u ε ,k ε  for fixed ε>0. Theorem 4.1. Under the assumptions of Theorem 3.6. Meyer problem P ε  has a solution. Proof. Let ε>0befixed.SinceJ ε w ≥ 0, there exists inf{J ε w,w ∈ W}.Denotem ε ≡ inf{J ε w,w ∈ W} and choose {w n }⊆W such that J ε w n  → m ε where w n u n ,k n  ∈ W  V ad × 0,  T. By assumption U, there exists a subsequence {u n }⊆V ad such that u n w → u ε in V ad as n →∞,andV ad is closed and convex, thanks to Mazur Lemma, u ε ∈ V ad .By assumption B ,wehave Bu n s −→ Bu ε , in L p  0, 1  ,X  , as n −→ ∞ . 4.1 Since k n k q n  is bounded and k n k q n  > 0, there also exists a subsequence {k n }{k q n } denoted by {k n }{k q n } ⊆ 0,  T again, such that k n  k q n  −→ k ε  k q ε  , in  0,  T  , as n −→ ∞ . 4.2 Let x n and x ε be the mild solutions of system 3.14 corresponding to w n u n ,k n  ∈ W and w ε u ε ,k ε  ∈ W, respectively. Then, we have x n  s   T n  s  z 0   s 0  s − θ  q−1 S n  s − θ  k q n F n  θ  dθ, x ε  s   T ε  s  z 0   s 0  s − θ  q−1 S ε  s − θ  k q ε F ε  θ  dθ, 4.3 Advances in Difference Equations 9 where T n  ·  ≡  ∞ 0 ξ q  θ  T k q n  · q θ  dθ, S n  ·  ≡ q  ∞ 0 θξ q  θ  T k q n  · q θ  dθ, F n  ·  ≡ f  k n ·,x n  ·  ,B  k n ·  u n  ·  , T ε  ·  ≡  ∞ 0 ξ q  θ  T k q ε  · q θ  dθ, S ε  ·  ≡ q  ∞ 0 θξ q  θ  T k q ε  · q θ  dθ, F ε  ·  ≡ f  k ε ·,x ε  ·  ,B  k ε ·  u ε  ·  . 4.4 By Lemma 3.7, assumptions F, B, U, and singular version Gronwall Lemma, it is easy to verify that there exists a constant ρ>0suchthat  x ε  C0,1,X ≤ ρ,  x n  C0,1,X ≤ ρ. 4.5 Further, there exists a constant M ε > 0suchthat  F ε  C0,1,X ≤ M ε  1  ρ   B  ∞ max t∈0,1 { u  t }  . 4.6 Denote R 1   T n  s  z 0 −T ε  s  z 0  , R 2       s 0  s − θ  q−1 S n  s − θ  k q n F n  θ  dθ −  s 0  s − θ  q−1 S n  s − θ  k q n F ε n  θ  dθ     , R 3       s 0  s − θ  q−1 S n  s − θ  k q n F ε n  θ  dθ −  s 0  s − θ  q−1 S ε  s − θ  k q ε F ε  θ  dθ     , 4.7 where F ε n  θ  ≡ f  k n θ, x ε  θ  ,B  k ε θ  u ε  θ  . 4.8 10 Advances in Difference Equations By assumption F, R 2 ≤ qC k q n k q n Γ  1  q   s 0  s − θ  q−1  F n  θ  − F ε n  θ  dθ ≤ qC k q n k q n L  ρ  Γ  1  q   s 0  s − θ  q−1  x n  θ  − x ε  θ  dθ  qC k q n k q n L  ρ  Γ  1  q   s 0  s − θ  q−1  B  k n θ  u n  θ  − B  k ε θ  u ε  θ  dθ ≤ R 21  R 22  R 23 , 4.9 where M k q n ≡ qC k q n k q n L  ρ  Γ  1  q  , R 21 ≡ M k q n  s 0  s − θ  q−1  x n  θ  − x ε  θ  dθ, R 22 ≡ M k q n  s 0  s − θ  q−1  B  k n θ  u ε  θ  − B  k ε θ  u ε  θ  dθ, R 23 ≡ M k q n  s 0  s − θ  q−1  B  k n θ  u n  θ  − B  k n θ  u ε  θ  dθ, R 3 ≤  s 0  s − θ  q−1    k q n S n  s − θ  F ε n  θ  − k q ε S n  s − θ  F ε  θ     dθ  k q ε  s 0  s − θ  q−1  S n  s − θ  F ε  θ  −S ε  s − θ  F ε  θ  dθ ≤ R 31  R 32  R 33 , 4.10 where R 31 ≡ M k q n k q n  s 0  s − θ  q−1  F ε n  θ  − F ε  θ  dθ, R 32 ≡ M k q n  s 0  s − θ  q−1    k q n − k q ε     F ε  θ  dθ, R 33 ≡ k q ε M ε  1  ρ   s 0  s − θ  q−1  S n  s − θ  −S ε  s − θ  dθ. 4.11 [...]... have x 1 z1 12 Advances in Difference Equations For any ε > 0, submitting w to Jε , we have 1 x wε 1 − z1 2ε τ ≥ J ε wε Jε w 2 kε 5.2 This inequality implies that 0 ≤ kε ≤ τ, 2 x wε 1 − z1 ≤ 2ετ, 5.3 hold for all ε > 0 We can choose a subsequence {εn } such that εn → 0 as n → ∞ and q kεn −→ kq 0 , kεn −→ k0 , in 0, T , in 0, T , in X, as n −→ ∞, x wεn 1 ≡ xεn 1 −→ z1 , w uεn −→ u0 , in Vad , 5.4 wεn... θ dθ, Advances in Difference Equations 13 where ∞ Tε n · ≡ 0 ∞ Sεn · ≡ q 0 q ξq θ Tkεn ·q θ dθ, q θξq θ Tkεn ·q θ dθ, Fεn · ≡ f kεn ·, xεn · , B kεn · uεn · , 5.7 ∞ T0 · ≡ ξq θ T k0 0 S0 · ≡ q · θ dθ, q q ∞ 0 θξq θ T k0 q ·q θ dθ, F 0 · ≡ f k0 ·, x0 · , B k0 · u0 · Recalling 5.5 and the process in Theorem 4.1, after some calculation, using the singular version Gronwall Lemma again, we also obtain s.. .Advances in Difference Equations 11 Note that Lemma 3.7 and 4.1 , combining Holder inequality with Lebesgue domi¨ nated convergence theorem, one can verify R1 → 0, R23 → 0, R31 → 0 and R33 → 0 q q as n → ∞ immediately Since kn kn → kε kε as n → ∞, Fε C 0,1 ,X and uε t E are bounded, R22 → 0, R32 → 0 as n → ∞ Then, we obtain that xn s − xε s ≤ R1 R2 ≤ σε q Mkn R3... evolution equations, ” Nonlinear Analysis: Theory, Methods & Applications, vol 71, no 10, pp 4471–4475, 2009 6 K Balachandran, S Kiruthika, and J J Trujillo, “Existence results for fractional impulsive integrodifferential equations in Banach spaces,” Communications in Nonlinear Science and Numerical Simulation, vol 16, no 4, pp 1970–1977, 2011 7 K Balachandran and J Y Park, “Controllability of fractional 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Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 385324, 16 pages doi:10.1155/2011/385324 Research Article Study of an Approximation. ∞ 3.19 uniformly in t on some closed interval of 0,  T in the strong operator topology sense. For system 3.14, we turn to consider the following Meyer problem. 8AdvancesinDifference Equations Meyer. fractional evolution equations in infinite dimensional spaces attract many authors including us see, for instance, 5–21 and the references therein. When the fractional differential equations describe